I am trying to prove the following:
Let $I, J$ be ideals in a ring $R$. Prove that the residue of any element of $I\cap J$ in $R/IJ$ is nilpotent.
I am not sure that I have got it right, here is my reasoning:
For any $x\in I\cap J$ we have $x^2\in IJ$. So $x^2+IJ=IJ$. This implies that $x^2=0$ in $R/IJ$.
Is this correct?